A Chebyshev collocation method for solving Symm's integral equation for conformal mapping: a partial error analysis

Levesley, Jeremy; Hough, D. M.; Chandler-Wilde, Simon N.

doi:10.1093/imanum/14.1.57

Cited by 6 publications

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“…This means that the truncated series (30) converges uniformly for any smooth boundary data and for all locations x near the boundary. This is a direct consequence of the excellent convergence properties of Chebyshev approximation applied to (24); a proof can be found in [33]. This property does not necessarily hold for other harmonic bases, in particular spherical harmonics, for which near-boundary convergence cannot be achieved, leading to a strong Gibbs effect.…”

Section: Exterior Harmonic Functionsmentioning

confidence: 99%

“…More general conformal mappings are often computed by solving Symm's or Carleman's equation (24) numerically on the domain boundary, making this approach similar in terms of numerical cost and precision to boundary integral equation methods. It is interesting to note that, for domains including corners, the Chebyshev approximation is especially well suited, guaranteeing superconvergence of the mapping function [33].…”

Section: Comparison With Other Approachesmentioning

confidence: 99%

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Spectral method for matching exterior and interior elliptic problems

Boronski

2007

Journal of Computational Physics

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A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green's function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

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Section: Exterior Harmonic Functionsmentioning

confidence: 99%

Section: Comparison With Other Approachesmentioning

confidence: 99%

Spectral method for matching exterior and interior elliptic problems

Boronski

2007

Journal of Computational Physics

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“…A similar strategy of using the Chebyshev points of the first kind can also be applied to other singular integral equations, such as integral equation of Symm's type; see e.g. [58,36,41].…”

Section: Solution Of Integral Equationsmentioning

confidence: 99%

The Chebyshev points of the first kind

2016

Applied Numerical Mathematics

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“…It is very natural in the case of an integral equation on an open arc, where the use of the cosine transformation (see [22], [33]) reduces the unknown to an infinitely smooth function if the data is smooth. This is the starting point for the (global) Chebyshev collocation method for (1.1) on an open arc -see [18], [19], [31]. However as is observed in [19] the extension of the analysis of this method to the case of a polygon is far from trivial.…”

Section: Introductionmentioning

confidence: 99%

“…This is the starting point for the (global) Chebyshev collocation method for (1.1) on an open arc -see [18], [19], [31]. However as is observed in [19] the extension of the analysis of this method to the case of a polygon is far from trivial. Change of variable techniques for other sorts of integral equations are considered, for example, in [25], [23], [17] and [3].…”

Section: Introductionmentioning

confidence: 99%

An optimal order collocation method for first kind boundary integral equations on polygons

Elschner

Graham

1995

Numerische Mathematik

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This paper discusses the convergence of the collocation method using splines of any order k for first kind integral equations with logarithmic kernels on closed polygonal boundaries in R 2 . Before discretization the equation is transformed to an equivalent equation over [−π, π] using a nonlinear parametrization of the polygon which varies more slowly than arc-length near each corner. This has the effect of producing a transformed equation with a solution which is smooth on [−π, π]. This latter integral equation is shown to be well-posed in appropriate Sobolev spaces. The structure of the integral operator is described in detail, and can be written in terms of certain non-standard Mellin convolution operators. Using this information we are able to show that the collocation method using splines of order k (degree k − 1) converges with optimal order O(h k ). (The collocation points are the mid-points of subintervals when k is odd and the break-points when k is even, and stability is shown under the assumption that the method may be modified slightly.) Using the numerical solutions to the transformed equation we construct numerical solutions of the original equation which converge optimally in a certain weighted norm. Finally the method is shown to produce superconvergent approximations to interior potentials such as those used to solve harmonic boundary value problems by the boundary integral method. The convergence results are illustrated with some numerical examples. Subject Classification (1991): 65R20, 45B05 Mathematics

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A Chebyshev collocation method for solving Symm's integral equation for conformal mapping: a partial error analysis

Cited by 6 publications

References 0 publications

Spectral method for matching exterior and interior elliptic problems

Spectral method for matching exterior and interior elliptic problems

The Chebyshev points of the first kind

An optimal order collocation method for first kind boundary integral equations on polygons

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